On the image-orthogonal exponentials Original Research Article

Let μM,DμM,D be a self-affine measure corresponding to a given affine iterated function system {ϕd(x)=M−1(x+d)}d∈D{ϕd(x)=M−1(x+d)}d∈D. In the present paper we will study the problem of how to determine the L2(μM,D)L2(μM,D)-space has finite or infinite orthogonal exponentials. Such research is motivated by a conjecture on the non-spectrality of μM,DμM,D. We first obtain a non-spectral criterion which extends the result of Dutkay and Jorgensen. In opposition to the condition of this criterion, we then obtain some conditions which imply the infinite orthogonal systems in L2(μM,D)L2(μM,D). These are necessary for further investigation on the spectrality of μM,DμM,D. As an application, we completely settle the corresponding problem for the generalized planar Sierpinski family.